# Problem evaluating a very tiny integral

Hi everyone,

I'm pretty new with sage. I was forced to change from MATLAB to Sage, because I was told Sage does approximate very tiny numbers better as it can work with sqrt(2) as sqrt(2) and not as the rational number approximating it.

Approximations are very important for my problem. I need to evaluate this integral $$\sum_{c=1}^{d}\int_{\min(256c-0.5,\frac{y(y+1)}{2})}^{\min(256c+0.5,\frac{y(y+1)}{2})}\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}\, dx$$

Suppose d = 1, then this is simply the integral

$$ \int_{255.5}^{256.5}\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}\, dx$$

To evaluate this integral I wrote the following code

T = RealDistribution('gaussian,1) print T.cum_distribution_function(256.5)-T.cum_distribution_function(255.5)

because the integral above is the same as the difference of the distribution function of a standard distributed gaussian random variable between the boundaries of the integral. However, and you can check yourselves if you don't believe it, the result I get with sage is 0.

I guess that this is due to some approximation, which sage does. Indeed the exact value of the integral is pretty close (and with pretty I mean a lot) to 0. My problem is that I need to be able to have the exact value, because the sum of integrals I'm working with and my whole work behind this integral requires me to be very careful with those little tiny numbers.

I tried to use the function

integrate

to deal with this problem, but something funny and apparently inexplicable happened when I was trying to use it.

To be precise I defined this code:

```
def VarianceOfvy(y):
temp1 = 0
temp2 = 0
for r in range(0,y+1):
for x in range(0,r+1):
temp1 = temp1 + (255/256)^x * 1/256 * (r-x)^2
for r in range(0,y+1):
for x in range(0,r+1):
temp2 = temp2 + ((255/256)^x * 1/256 * (r-x))^2
sigma = temp1 - temp2
return sqrt(sigma)
def Integerd(y):
b = y*(y+1)/2
d = 1
c = 0
while min((c+1)*256-0.5,b) == (c+1)*256-0.5:
d = d+1
c = c+1
return d
def Probabilityvynequiv(y):
var('c')
b = (y*(y+1))/2
sigma = 2
mu = 1
d = Integerd(y)
factor = 1/(integrate(1/sqrt(2*pi)*e^(-(x/sigma)^2/2),x,-oo,(b-mu)/sigma) - integrate(1/sqrt(2*pi)*e^(-(x)^2/2),x,-oo,(-mu)/sigma))
p = sum(factor*1/sigma*integrate(1/sqrt(2*pi)*e^((-x^2)/(2)),x,c*256+0.5,min((c+1)*256-0.5,b)),c,0,d)
return p
```

And if I let it run, the result I get is

```
1/2*(erf(255.75*sqrt(2)) - erf(128.25*sqrt(2)) + erf(127.75*sqrt(2)) - erf(0.25*sqrt(2)))/(3*erf(1/4*sqrt(2)) + 1)
```

which I assume is correct, and at least it tells me that Sage is able to read my code and output a result. If I call the function VarianceOfvy(2), the result I get is 3/65536*sqrt(11133895), which is also correct. Now, if I'm changing the command

```
sigma = 2
```

with

```
sigma = VarianceOfvy(2)
```

and try to let the whole program run again, Sage is not able anymore to output a result.

I'm really lost and I don't know what to do. Could someone advise me and give me some hints on how to evaluate those tiny integrals, in such a way that I don't loose any approximation?

You can enclose the LaTeX code in

`$`

(or`$$`

) to make it readable.The error function is in Sage and can be useful.

@mforets I don't know what you mean. If you tell Sage to compute the difference between erf(256.5) and erf(255.5), with the code print erf(256.5)-erf(255.5), it tells you that it is 0. Is there a way to make it more precise?

Ok, i see. Actually with 10000 bits of precision it's still not enough:

I don't know; the exponential of

`-x^2`

goes fast to zero, can we estimate how much precision is needed here? With one more order of magnitude of precision, i get someting non-zero (long time!):